where \(\mathcal {T}_{k+1}^{(j)}=\lbrace \underline{l} =(l_1,\ldots ,l_{k+1}) \in \mathcal {S}_{k+1}~|~ l_m \ne j\rbrace ,\)    \( j=0,\ldots ,n.\) 
For the
properties of \(\Psi _{k+1}(\bar{\mu })\)  and
\(\Phi _{k+1}^{(j)}(\bar{\mu })\) , we refer to Appendix E [1]}.
